In a large, shallow dish, combine 4 tbsp. EVOO, half the lemon juice, the lemon zest, garlic and thyme. Preheat a grill pan or griddle over medium-high heat.
Cut into the chicken breasts horizontally to butterfly, opening them up like a book. Firmly but gently pound them into very thin cutlets; season with salt and pepper. Add them to the lemon-garlic marinade and let stand for 5 minutes.
Meanwhile, in a medium skillet, melt the butter over medium heat. Whisk in the flour and cook until thickened; whisk in the chicken stock and season with salt and pepper. Whisk in the mustard, then stir in the tarragon. Keep the sauce warm over low heat.
On the grill pan or griddle, working in batches if necessary, grill the chicken, turning once, until cooked through, 4 to 6 minutes.
In a small bowl, whisk the remaining lemon juice, the vinegar and the remaining 2 tbsp. EVOO; season with salt and pepper. Add the arugula, peas, radishes and shallot and toss gently with your hands to lightly coat. Top with the cheese.
To serve, ladle some of the sauce onto plates. Top with the chicken. Pile the salad over the chicken.